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arXiv:hep-th/0308007

Quantum systems in a cut Fock space

Maciej Trzetrzelewski, J. Wosiek

Published Aug 1, 2003• 9 citations
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Abstract

Standard quantum mechanics is viewed as a limit of a cut system with artificially restricted dimension of a Hilbert space. Exact spectrum of cut momentum and coordinate operators is derived and the limiting transition to the infinite dimensional Hilbert space is studied in detail. The difference between systems with discrete and continuous energy spectra is emphasized. In particular a new scaling law, characteristic for nonlocalized, states is found. Some applications for supersymmetric quantum mechanics are briefly outlined.

DOI
10.48550/arxiv.hep-th/0308007

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Standard quantum mechanics is viewed as a limit of a cut system with artificially restricted dimension of a Hilbert space. Exact spectrum of cut momentum and coordinate operators is derived and the limiting transition to the infinite dimensional Hilbert space is studied in detail. The difference between systems with discrete and continuous energy spectra is emphasized. In particular a new scaling law, characteristic for nonlocalized, states is found. Some applications for supersymmetric quantum mechanics are briefly outlined.
This paper contributes to a recent series discussing quantum mechanics defined on a finite-dimensional Hilbert space in which Weyl's commutation relation for unitary operators holds. In an earlier paper, Santhanam and Tekumalla (1976) showed that the commutation relation for hermitian operators with a bounded spectrum tends to Hdsenberg's standard canonical commutation relation as the spectrum becomes continuous and the dimension n -> 00. The present paper offers a formulation which is coordinate free in the limit n -> 00 and makes the limiting procedure especially ransparent.
We begin with a presentation of a minimal set of von Neumann postulates while introducing language and notation to facilitate subsequent discussion of quantum calculations based in finite-dimensional Hilbert spaces. We also address two-state quantum systems, entanglement of multiple two-state systems, quantum angular momentum theory and quantum approaches to statistical mechanics. A concluding chapter gives an overview of issues associated with quantum mechanics in continuous infinite-dimensional Hilbert spaces.
The complex Hilbert space of standard quantum mechanics may be treated as a real Hilbert space. The pure states of the complex theory become mixed states in the real formulation. It is then possible to generalize standard quantum mechanics, keeping the same set of physical states, but admitting more general observables. The standard time reversal operator involves complex conjugation, in this sense it goes beyond the complex theory and may serve as an example to motivate the generalization. Another example is unconventional canonical quantization such that the harmonic oscillator of angular frequency ωωω has any given finite or infinite set of discrete energy eigenvalues, limited below by ℏω/2\hbarω/2ℏω/2.
Classical mechanics is usually valid at the macro scale. The changes in the position and the momentum of the object over time are described by the Hamiltonian equation of motion. The wave function in quantum mechanics, if unobserved, evolves in a smooth and continuous way according to the Schrodinger equation, which is related to the Hamiltonian equation of motion. This equation describes a linear superposition of different states at time. A quantum mechanical description of a physical system is related to a probabilistic representation; it is described by a vector in Hilbert space. This description extends the two- or three-dimensional Euclidean space into spaces that have any finite or infinite number of dimensions. The state of a compound system is projected to the subspace that corresponds to the observed state and the vector representing the state is renormalized to the unit length. An observable describes a subspace for some dimensions with a special case of one dimension.
Quantum mechanics is formulated using a nonstandard Hilbert space. The concept of an eigen vector of a linear operator, which applies to standard as well as nonstandard Hilbert spaces, is replaced by the more general concept of an ultra eigen vector, which applies to nonstandard Hilbert spaces alone. Ultra eigen vectors corresponding to all spectral points of internal self−adjoint operators are proved to exist. This result enables us to set up a formalism which is equally valid for the discrete, as well as the continuous spectrum. Finally, Dirac’s formalism is reproduced, in a rigorous form within the nonstandard Hilbert space structure.